Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x+8y &= 2 \\ 4x+9y &= 5\end{align*}$
Explanation: Begin by moving the $x$ -term in the second equation to the right side of the equation. $9y = -4x+5$ Divide both sides by $9$ to isolate $y$ $y = {-\dfrac{4}{9}x + \dfrac{5}{9}}$ Substitute this expression for $y$ in the first equation. $-5x+8({-\dfrac{4}{9}x + \dfrac{5}{9}}) = 2$ $-5x - \dfrac{32}{9}x + \dfrac{40}{9} = 2$ Simplify by combining terms, then solve for $x$ $-\dfrac{77}{9}x + \dfrac{40}{9} = 2$ $-\dfrac{77}{9}x = -\dfrac{22}{9}$ $x = \dfrac{2}{7}$ Substitute $\dfrac{2}{7}$ for $x$ back into the top equation. $-5( \dfrac{2}{7})+8y = 2$ $-\dfrac{10}{7}+8y = 2$ $8y = \dfrac{24}{7}$ $y = \dfrac{3}{7}$ The solution is $\enspace x = \dfrac{2}{7}, \enspace y = \dfrac{3}{7}$.